Monatsh Math (2018) 186:215–233
Orbit Dirichlet series and multiset permutations
· Christopher Voll
Received: 4 January 2017 / Accepted: 25 October 2017 / Published online: 4 November 2017
© Springer-Verlag GmbH Austria 2017
Abstract We study Dirichlet series enumerating orbits of Cartesian products of maps
whose orbit distributions are modelled on the distributions of ﬁnite index subgroups
of free abelian groups of ﬁnite rank. We interpret Euler factors of such orbit Dirichlet
series in terms of generating polynomials for statistics on multiset permutations, viz.
descent and major index, generalizing Carlitz’s q-Eulerian polynomials. We give two
main applications of this combinatorial interpretation. Firstly, we establish local func-
tional equations for the Euler factors of the orbit Dirichlet series under consideration.
Secondly, we determine these (global) Dirichlet series’ abscissae of convergence and
establish some meromorphic continuation beyond these abscissae. As a corollary, we
describe the asymptotics of the relevant orbit growth sequences. For Cartesian products
of more than two maps we establish a natural boundary for meromorphic continuation.
For products of two maps, we prove the existence of such a natural boundary subject
to a combinatorial conjecture.
Keywords Orbit Dirichlet series · Multiset permutations · Carlitz’s q-Eulerian
polynomials · Hadamard products of rational generating functions · Igusa functions ·
local functional equations · Natural boundaries
Mathematics Subject Classiﬁcation 37C30 · 37P35 · 30B50 · 11M41 · 05A15 ·
Communicated by A. Constantin.
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany